4 SIMPLE ARCS AND SIMPLE CLOSED CURVES 259

homeomorphism/i (resp./2) of UA (resp. U2) onto the closed interval J = [y, z\
such that/^) =f2(eic) = y,ft(eidy=f2(eid) = z. Let hą (resp. h2) be the map-
ping of U into C, equal to/in U^ (resp. in U2), to/2 in U2 (resp. tof^ in U^);
the definition of J implies that Al5 h2 are homeomorphisms of U onto two
simple closed curves Gx = H^ u J, G2 = H2 u J, each of which contains the
segment J. Let w e Hl9 distinct of y and z; there is an open ball D of center w,
which does not meet the compact set G2. From (Ap.4.2.1), each connected
component of C — Gx has points in D; moreover, if wf, w" are two points of
D in a same connected component of C — Gl9 w' and w" are not separated
by Gx; they are not separated either by G2, since they belong to D c C — G2
which is connected. But G1 n G2 = J is connected, hence, by Janiszewski's
theorem (Ap.3.2), w' and w" are not separated by G1 u G2, nor of course
by H c Gx u G2. In other words, w' and w" belong to the same connected
component of C — H. But as C — Gx has exactly two connected components,
and each connected component of C — H has points in D by (Ap.4.2.1), it
follows that C — H has at most two connected components. On the other
hand, it follows from (Ap.4.2.2) that there are two points w', w" in D which
are separated by Gx. We show they are separated by H. Otherwise, as they
are not separated by G2, and G2 n H = H2 is connected, they would not be
separated by G2 u H z> Gi (Ap.3.2), contrary to assumption. We have thus
shown that C — H has exactly two connected components; the same argument
as in (Ap.4.2.2) proves that one of them, A, is unbounded and the other, B, is
bounded.

Finally, we can suppose y is the origin of the loop y, and, if I = [a, jj],
that H! = y([a, A]), H2 = y([l, /?]). Define the loops y1 and y2 as follows:
yi(t) = (t - a H- l)(y - z) + z for a - 1 < t < a, y^t) = y(t) for a < t < A;
72(0 = 7(0 for !<*<Ģ/?, y2(f) = y + (t - /J)(z -/} for j3<f</T+l.
Using (Ap.1.1) it is immediately verified that for any point X$G! u G2,
j(x; y) = j(x; yt) + j(x; y2). With the same meaning as above for D, let again
w', w" be two points of D separated by Gą; then we have j(w'; y2) = j(w"; y2)
since w' and w" are not separated by G2 (9.8.3), and j(w'; y^) — j(w"; yt) = ą1
by (Ap.4.2.2). From this it follows that j(w'; y) - j(w"; y) = ą1, which ends
the proof.

(Ap.4.3) Let H be a simple closed curve in C, D the bounded connected
component of C — H. Then, for any loop y in D, j(x; y) = 0 for any xeH.

Let U be an open ball of center x, having no common points with the set
y(I) of points of y. There exists in U a point zeC-(DuH) = C-B
(Ap.4.2), and as U is connected, j(x; y) = j(z; y) (9.8.3). But j(z\ y) = j(y; y)y1 be the path f->y(0 defined in J, of